The data set contains 1989 records. The overall descriptive statistics:

##     Married     Meet credit history guidelines
##  No     : 678   0  : 171                      
##  Unknown:   3   1  :1816                      
##  Yes    :1308   666:   2                      
##                                               
##                                               
##                                               
##  Other obligations as a percent of total income non-Hispanic Black
##  Min.   : 0.00                                  No :1792          
##  1st Qu.:28.00                                  Yes: 197          
##  Median :33.00                                                    
##  Mean   :32.39                                                    
##  3rd Qu.:37.00                                                    
##  Max.   :95.00                                                    
##  Hispanic        Male      Mortgage loan approved
##  No :1878   No     : 369   No : 244              
##  Yes: 111   Unknown:  15   Yes:1745              
##             Yes    :1605                         
##                                                  
##                                                  
##                                                  
##  Loan amount/purchase price                 Race     
##  Min.   :  2.105            Hispanic          : 111  
##  1st Qu.: 70.000            non-Hispanic Black: 197  
##  Median : 80.000            non-Hispanic White:1681  
##  Mean   : 77.064                                     
##  3rd Qu.: 89.894                                     
##  Max.   :257.143

Descriptive statistics by Race:

## $Hispanic
##     MARRIED   GDLIN        OBRAT            MALE    APPROVE 
##  No     :31   0  :16   Min.   :14.60   No     :22   No :26  
##  Unknown: 1   1  :95   1st Qu.:29.00   Unknown: 2   Yes:85  
##  Yes    :79   666: 0   Median :33.00   Yes    :87           
##                        Mean   :33.46                        
##                        3rd Qu.:38.45                        
##                        Max.   :62.00                        
##     LOANPRC                       RACE    
##  Min.   : 39.39   Hispanic          :111  
##  1st Qu.: 80.00   non-Hispanic Black:  0  
##  Median : 89.39   non-Hispanic White:  0  
##  Mean   : 85.17                           
##  3rd Qu.: 90.42                           
##  Max.   :162.63                           
## 
## $`non-Hispanic Black`
##     MARRIED    GDLIN         OBRAT            MALE     APPROVE  
##  No     : 76   0  : 53   Min.   : 5.60   No     : 51   No : 64  
##  Unknown:  0   1  :144   1st Qu.:31.00   Unknown:  2   Yes:133  
##  Yes    :121   666:  0   Median :35.00   Yes    :144            
##                          Mean   :34.94                          
##                          3rd Qu.:38.90                          
##                          Max.   :63.00                          
##     LOANPRC                       RACE    
##  Min.   : 28.99   Hispanic          :  0  
##  1st Qu.: 80.00   non-Hispanic Black:197  
##  Median : 87.02   non-Hispanic White:  0  
##  Mean   : 83.97                           
##  3rd Qu.: 90.24                           
##  Max.   :255.52                           
## 
## $`non-Hispanic White`
##     MARRIED     GDLIN          OBRAT            MALE      APPROVE   
##  No     : 571   0  : 102   Min.   : 0.00   No     : 296   No : 154  
##  Unknown:   2   1  :1577   1st Qu.:27.60   Unknown:  11   Yes:1527  
##  Yes    :1108   666:   2   Median :32.50   Yes    :1374             
##                            Mean   :32.02                            
##                            3rd Qu.:36.50                            
##                            Max.   :95.00                            
##     LOANPRC                        RACE     
##  Min.   :  2.105   Hispanic          :   0  
##  1st Qu.: 68.182   non-Hispanic Black:   0  
##  Median : 79.888   non-Hispanic White:1681  
##  Mean   : 75.719                            
##  3rd Qu.: 89.623                            
##  Max.   :257.143

Descriptive statistics by Marital Status:

## $No
##     MARRIED    GDLIN         OBRAT            MALE     APPROVE  
##  No     :678   0  : 64   Min.   : 4.00   No     :252   No :102  
##  Unknown:  0   1  :614   1st Qu.:28.00   Unknown:  7   Yes:576  
##  Yes    :  0   666:  0   Median :33.00   Yes    :419            
##                          Mean   :32.74                          
##                          3rd Qu.:37.00                          
##                          Max.   :83.00                          
##     LOANPRC                        RACE    
##  Min.   :  2.105   Hispanic          : 31  
##  1st Qu.: 72.426   non-Hispanic Black: 76  
##  Median : 80.000   non-Hispanic White:571  
##  Mean   : 77.967                           
##  3rd Qu.: 89.978                           
##  Max.   :162.626                           
## 
## $Unknown
##     MARRIED  GDLIN       OBRAT           MALE   APPROVE    LOANPRC      
##  No     :0   0  :0   Min.   :13.0   No     :1   No :0   Min.   : 86.96  
##  Unknown:3   1  :3   1st Qu.:23.3   Unknown:0   Yes:3   1st Qu.: 88.62  
##  Yes    :0   666:0   Median :33.6   Yes    :2           Median : 90.29  
##                      Mean   :27.2                       Mean   : 98.16  
##                      3rd Qu.:34.3                       3rd Qu.:103.76  
##                      Max.   :35.0                       Max.   :117.24  
##                  RACE  
##  Hispanic          :1  
##  non-Hispanic Black:0  
##  non-Hispanic White:2  
##                        
##                        
##                        
## 
## $Yes
##     MARRIED     GDLIN          OBRAT            MALE      APPROVE   
##  No     :   0   0  : 107   Min.   : 0.00   No     : 116   No : 142  
##  Unknown:   0   1  :1199   1st Qu.:28.00   Unknown:   8   Yes:1166  
##  Yes    :1308   666:   2   Median :33.00   Yes    :1184             
##                            Mean   :32.22                            
##                            3rd Qu.:37.00                            
##                            Max.   :95.00                            
##     LOANPRC                        RACE     
##  Min.   :  8.772   Hispanic          :  79  
##  1st Qu.: 68.857   non-Hispanic Black: 121  
##  Median : 80.000   non-Hispanic White:1108  
##  Mean   : 76.547                            
##  3rd Qu.: 89.866                            
##  Max.   :257.143

Descriptive statistics by Gender:

## $No
##     MARRIED    GDLIN         OBRAT            MALE     APPROVE  
##  No     :252   0  : 31   Min.   : 6.99   No     :369   No : 50  
##  Unknown:  1   1  :338   1st Qu.:28.00   Unknown:  0   Yes:319  
##  Yes    :116   666:  0   Median :33.00   Yes    :  0            
##                          Mean   :32.64                          
##                          3rd Qu.:37.00                          
##                          Max.   :83.00                          
##     LOANPRC                       RACE    
##  Min.   : 11.01   Hispanic          : 22  
##  1st Qu.: 70.83   non-Hispanic Black: 51  
##  Median : 80.00   non-Hispanic White:296  
##  Mean   : 77.66                           
##  3rd Qu.: 90.00                           
##  Max.   :255.52                           
## 
## $Unknown
##     MARRIED  GDLIN        OBRAT            MALE    APPROVE 
##  No     :7   0  : 0   Min.   :24.00   No     : 0   No : 0  
##  Unknown:0   1  :15   1st Qu.:29.95   Unknown:15   Yes:15  
##  Yes    :8   666: 0   Median :34.50   Yes    : 0           
##                       Mean   :33.33                        
##                       3rd Qu.:37.65                        
##                       Max.   :40.30                        
##     LOANPRC                      RACE   
##  Min.   :39.39   Hispanic          : 2  
##  1st Qu.:74.93   non-Hispanic Black: 2  
##  Median :75.42   non-Hispanic White:11  
##  Mean   :75.59                          
##  3rd Qu.:80.43                          
##  Max.   :92.90                          
## 
## $Yes
##     MARRIED     GDLIN          OBRAT            MALE      APPROVE   
##  No     : 419   0  : 140   Min.   : 0.00   No     :   0   No : 194  
##  Unknown:   2   1  :1463   1st Qu.:28.00   Unknown:   0   Yes:1411  
##  Yes    :1184   666:   2   Median :33.00   Yes    :1605             
##                            Mean   :32.32                            
##                            3rd Qu.:37.00                            
##                            Max.   :95.00                            
##     LOANPRC                        RACE     
##  Min.   :  2.105   Hispanic          :  87  
##  1st Qu.: 69.655   non-Hispanic Black: 144  
##  Median : 80.000   non-Hispanic White:1374  
##  Mean   : 76.942                            
##  3rd Qu.: 89.881                            
##  Max.   :257.143

There are 3 records are missing married (MARRIED) field.

ID MARRIED GDLIN OBRAT BLACK HISPAN MALE APPROVE LOANPRC
356 Unknown 1 35.0 No Yes Yes Yes 86.95652
759 Unknown 1 33.6 No No Yes Yes 90.28571
1392 Unknown 1 13.0 No No No Yes 117.24140

There are 3 records are missing married (GDLIN) field.

ID MARRIED GDLIN OBRAT BLACK HISPAN MALE APPROVE LOANPRC
881 Yes 666 35 No No Yes Yes 75.82939
1229 Yes 666 26 No No Yes Yes 100.00000

There are 15 records are missing gender (MALE) field.

ID MARRIED GDLIN OBRAT BLACK HISPAN MALE APPROVE LOANPRC
1 No 1 34.5 No No Unknown Yes 75.42373
127 No 1 31.6 No No Unknown Yes 80.80000
286 Yes 1 37.3 No No Unknown Yes 80.05337
452 Yes 1 40.1 Yes No Unknown Yes 75.00000
618 Yes 1 38.5 No No Unknown Yes 92.90323
695 Yes 1 25.0 No No Unknown Yes 64.48276
762 Yes 1 27.6 No No Unknown Yes 75.55556
768 No 1 35.6 No No Unknown Yes 64.74397
833 Yes 1 24.0 No Yes Unknown Yes 79.80769
979 No 1 31.7 No No Unknown Yes 74.86033
1040 No 1 38.0 Yes No Unknown Yes 75.38462
1070 Yes 1 40.3 No Yes Unknown Yes 39.39394
1092 Yes 1 29.7 No No Unknown Yes 90.10239
1613 No 1 30.2 No No Unknown Yes 90.00000
1924 No 1 35.8 No No Unknown Yes 75.32051

There are 33 records (98.3408748 %) that have LOANPRC > 100%. There are a total of 1989 records.

ID MARRIED GDLIN OBRAT BLACK HISPAN MALE APPROVE LOANPRC
9 Yes 1 30.7 No No Yes Yes 100.3009
37 Yes 1 35.0 No No Yes No 112.7119
53 Yes 1 38.0 No No Yes Yes 108.5714
132 Yes 1 27.6 No No Yes Yes 101.1905
162 Yes 1 25.0 No No Yes Yes 164.7059
233 Yes 1 18.0 No No Yes Yes 105.5556
258 Yes 1 45.0 No No Yes No 220.0000
262 Yes 0 56.0 No No Yes No 257.1429
294 Yes 1 37.0 No No Yes Yes 148.8095
490 Yes 1 25.0 Yes No Yes Yes 120.5480
544 Yes 0 52.0 No No Yes Yes 117.6471
560 Yes 1 33.0 No Yes No No 137.9310
565 No 1 10.0 No No Yes Yes 133.3333
750 Yes 1 35.0 No No Yes No 147.8261
793 Yes 1 37.0 No No Yes No 130.0000
798 No 1 36.0 No No No Yes 147.3684
841 No 1 41.2 No No Yes No 100.0676
904 Yes 1 24.0 No No Yes Yes 183.3333
914 Yes 1 40.0 No Yes Yes Yes 111.4286
940 Yes 1 38.0 No No Yes Yes 128.9474
947 Yes 1 18.0 Yes No Yes Yes 100.3521
963 Yes 0 41.3 No Yes Yes No 109.4118
1112 No 0 35.5 No Yes No No 162.6263
1206 No 1 41.0 No No No Yes 125.0000
1322 No 1 32.0 No No Yes Yes 110.5263
1392 Unknown 1 13.0 No No No Yes 117.2414
1394 Yes 1 37.0 No No Yes Yes 114.0845
1457 Yes 1 19.0 No No Yes Yes 108.0000
1623 Yes 1 29.0 No No Yes Yes 109.0909
1632 Yes 1 36.8 Yes No No Yes 255.5248
1667 No 1 43.0 No No Yes Yes 120.0000
1683 No 1 25.0 No No No Yes 118.7500
1759 No 1 46.0 No No No Yes 107.2000
## $Hispanic
##     MARRIED   GDLIN        OBRAT            MALE    APPROVE 
##  No     :30   0  :14   Min.   :14.60   No     :20   No :23  
##  Unknown: 1   1  :93   1st Qu.:29.00   Unknown: 2   Yes:84  
##  Yes    :76   666: 0   Median :33.00   Yes    :85           
##                        Mean   :33.31                        
##                        3rd Qu.:38.10                        
##                        Max.   :62.00                        
##     LOANPRC                       RACE    
##  Min.   : 39.39   Hispanic          :107  
##  1st Qu.: 80.00   non-Hispanic Black:  0  
##  Median : 88.46   non-Hispanic White:  0  
##  Mean   : 83.48                           
##  3rd Qu.: 90.23                           
##  Max.   :100.00                           
## 
## $`non-Hispanic Black`
##     MARRIED    GDLIN         OBRAT            MALE     APPROVE  
##  No     : 76   0  : 53   Min.   : 5.60   No     : 50   No : 64  
##  Unknown:  0   1  :141   1st Qu.:31.00   Unknown:  2   Yes:130  
##  Yes    :118   666:  0   Median :35.00   Yes    :142            
##                          Mean   :35.07                          
##                          3rd Qu.:38.90                          
##                          Max.   :63.00                          
##     LOANPRC                       RACE    
##  Min.   : 28.99   Hispanic          :  0  
##  1st Qu.: 79.96   non-Hispanic Black:194  
##  Median : 84.22   non-Hispanic White:  0  
##  Mean   : 82.81                           
##  3rd Qu.: 90.23                           
##  Max.   :100.00                           
## 
## $`non-Hispanic White`
##     MARRIED     GDLIN          OBRAT            MALE      APPROVE   
##  No     : 563   0  : 100   Min.   : 0.00   No     : 291   No : 148  
##  Unknown:   1   1  :1553   1st Qu.:27.65   Unknown:  11   Yes:1507  
##  Yes    :1091   666:   2   Median :32.50   Yes    :1353             
##                            Mean   :32.00                            
##                            3rd Qu.:36.40                            
##                            Max.   :95.00                            
##     LOANPRC                        RACE     
##  Min.   :  2.105   Hispanic          :   0  
##  1st Qu.: 67.797   non-Hispanic Black:   0  
##  Median : 79.861   non-Hispanic White:1655  
##  Mean   : 74.832                            
##  3rd Qu.: 89.237                            
##  Max.   :100.000
##                     APPROVE
## RACE                   No  Yes
##   Hispanic             26   85
##   non-Hispanic Black   64  133
##   non-Hispanic White  154 1527

##          APPROVE
## MARRIED     No  Yes
##   No       102  576
##   Unknown    0    3
##   Yes      142 1166

##          APPROVE
## MALE        No  Yes
##   No        50  319
##   Unknown    0   15
##   Yes      194 1411

## , , RACE = Hispanic
## 
##          APPROVE
## MARRIED     No  Yes
##   No         7   24
##   Unknown    0    1
##   Yes       19   60
## 
## , , RACE = non-Hispanic Black
## 
##          APPROVE
## MARRIED     No  Yes
##   No        27   49
##   Unknown    0    0
##   Yes       37   84
## 
## , , RACE = non-Hispanic White
## 
##          APPROVE
## MARRIED     No  Yes
##   No        68  503
##   Unknown    0    2
##   Yes       86 1022

Data Discription

The data set includes the following variables:

Logistic Regression Full Model

log(p/1-p) = \(b_0 + b_1 * GDLIN + b_2 * OBRAT + b_3 * BLACK + b_4 * HISPAN + b_5 * LOANPRC + b_6 * MARRIED + b_7 * MALE\)

## 
## Call:
## glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.8933   0.2445   0.3128   0.3742   2.3261  
## 
## Coefficients:
##              Estimate Std. Error z value             Pr(>|z|)    
## (Intercept)  1.381531   0.591656   2.335             0.019542 *  
## GDLIN1       3.719269   0.217169  17.126 < 0.0000000000000002 ***
## OBRAT       -0.034074   0.010310  -3.305             0.000950 ***
## BLACK1      -0.815693   0.240177  -3.396             0.000683 ***
## HISPAN1     -0.900010   0.310585  -2.898             0.003758 ** 
## LOANPRC     -0.016812   0.005074  -3.313             0.000922 ***
## MARRIED1     0.475742   0.192005   2.478             0.013221 *  
## MALE1       -0.053947   0.234573  -0.230             0.818107    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1475.4  on 1968  degrees of freedom
## Residual deviance:  959.4  on 1961  degrees of freedom
## AIC: 975.4
## 
## Number of Fisher Scoring iterations: 6

For every one unit change in OBRAT, the log odds of loan approval (versus non loan approval) decreases by 0.0340739.

For every one unit change in LOANPRC, the log odds of loan approval (versus non loan approval) decreases by 0.0168119.

The log odds of loan approval for applicants that meet credit guidelines increases by 3.719269.

The log odds of loan approval for married applicants increases by 0.4757419.

The log odds of loan approval for Black applicants decreases by 0.8156932.

The log odds of loan approval for Hispanic applicants decreases by 0.9000102.

For example, for a black married male whose credit history meets guidline (GDLIN = 1), loan amount price is 100 (LOANPRC = 100) and other obligations as a percent of total income is none (OBRAT = 0), the log odds of loan approval is 95.37%

Statistical Tests for Individual Predictors

##            Overall
## GDLIN1   17.126161
## OBRAT     3.305071
## BLACK1    3.396220
## HISPAN1   2.897795
## LOANPRC   3.313389
## MARRIED1  2.477753
## MALE1     0.229981
## Wald test for GDLIN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## F =  293.3054  on  1  and  1961  df: p= < 0.000000000000000222
## Wald test for OBRAT
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## F =  10.9235  on  1  and  1961  df: p= 0.00096664
## Wald test for BLACK
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## F =  11.53431  on  1  and  1961  df: p= 0.00069686
## Wald test for HISPAN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## F =  8.397215  on  1  and  1961  df: p= 0.0037997
## Wald test for MALE
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## F =  0.05289126  on  1  and  1961  df: p= 0.81813
## Wald test for LOANPRC
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## F =  10.97855  on  1  and  1961  df: p= 0.00093849
## Wald test for MARRIED
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = "binomial", data = data1)
## F =  6.13926  on  1  and  1961  df: p= 0.013305
##          llh      llhNull           G2     McFadden         r2ML 
## -479.7016510 -737.7148785  516.0264551    0.3497465    0.2305470 
##         r2CU 
##    0.4372079

Logistic Regression Reduced Model 1

log(p/1-p) = \(b_0 + b_1 * GDLIN + b_2 * OBRAT + b_3 * BLACK + b_4 * HISPAN + b_5 * LOANPRC + b_6 * MARRIED\)

## 
## Call:  glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## 
## Coefficients:
## (Intercept)       GDLIN1        OBRAT       BLACK1      HISPAN1  
##     1.34239      3.72139     -0.03410     -0.81143     -0.89733  
##     LOANPRC     MARRIED1  
##    -0.01677      0.46093  
## 
## Degrees of Freedom: 1968 Total (i.e. Null);  1962 Residual
## Null Deviance:       1475 
## Residual Deviance: 959.5     AIC: 973.5
## Wald test for GDLIN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## F =  294.1283  on  1  and  1962  df: p= < 0.000000000000000222
## Wald test for OBRAT
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## F =  10.93867  on  1  and  1962  df: p= 0.00095878
## Wald test for BLACK
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## F =  11.47424  on  1  and  1962  df: p= 0.00071961
## Wald test for HISPAN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## F =  8.358631  on  1  and  1962  df: p= 0.0038808
## Wald test for MARRIED
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## F =  6.482227  on  1  and  1962  df: p= 0.010972
## Wald test for LOANPRC
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## F =  10.92337  on  1  and  1962  df: p= 0.00096669

For every one unit change in OBRAT, the log odds of loan approval (versus non loan approval) decreases by 0.0340979.

For every one unit change in LOANPRC, the log odds of loan approval (versus non loan approval) decreases by 0.0167734.

The log odds of loan approval for applicants that meet credit guidelines increase by 3.721387.

The log odds of loan approval for married applicants decreases by 0.4609301.

The log odds of loan approval for non married applicants decreases by NA.

The log odds of loan approval for Black applicants decreases by 0.8114263.

The log odds of loan approval for Hispanic applicants decreases by 0.8973309.

For example, for a black married person whose credit history meets guidline (GDLIN = 1), loan amount price is 100 (LOANPRC = 100) and other obligations as a percent of total income is none (OBRAT = 0), the log odds of loan approval is 95.42%

CIs using profiled log-likelihood

##                   2.5 %       97.5 %
## (Intercept)  0.23664368  2.465735226
## GDLIN1       3.30547346  4.157682544
## OBRAT       -0.05434377 -0.013920887
## BLACK1      -1.27292218 -0.332543498
## HISPAN1     -1.48462721 -0.264593691
## LOANPRC     -0.02697832 -0.006937324
## MARRIED1     0.10487116  0.815796783

CIs using standard errors

##                   2.5 %       97.5 %
## (Intercept)  0.23158635  2.453187301
## GDLIN1       3.29609767  4.146676358
## OBRAT       -0.05430447 -0.013891283
## BLACK1      -1.28092587 -0.341926697
## HISPAN1     -1.50565235 -0.289009441
## LOANPRC     -0.02672039 -0.006826433
## MARRIED1     0.10609951  0.815760700

Odds ratios only

## (Intercept)      GDLIN1       OBRAT      BLACK1     HISPAN1     LOANPRC 
##   3.8281698  41.3216682   0.9664769   0.4442240   0.4076563   0.9833665 
##    MARRIED1 
##   1.5855480

Odds ratios and 95% CI

##                     OR      2.5 %     97.5 %
## (Intercept)  3.8281698  1.2669896 11.7721341
## GDLIN1      41.3216682 27.2614457 63.9232116
## OBRAT        0.9664769  0.9471065  0.9861756
## BLACK1       0.4442240  0.2800122  0.7170975
## HISPAN1      0.4076563  0.2265868  0.7675177
## LOANPRC      0.9833665  0.9733823  0.9930867
## MARRIED1     1.5855480  1.1105675  2.2609765

Logistic Regression Reduced Model 2

log(p/1-p) = \(b_0 + b_1 * GDLIN + b_2 * OBRAT + b_3 * BLACK + b_4 * HISPAN + b_5 * LOANPRC\)

## 
## Call:  glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = "binomial", data = data1)
## 
## Coefficients:
## (Intercept)       GDLIN1        OBRAT       BLACK1      HISPAN1  
##     1.66805      3.69461     -0.03512     -0.81768     -0.85828  
##     LOANPRC  
##    -0.01659  
## 
## Degrees of Freedom: 1968 Total (i.e. Null);  1963 Residual
## Null Deviance:       1475 
## Residual Deviance: 965.9     AIC: 977.9
## Wald test for GDLIN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = "binomial", data = data1)
## F =  295.2972  on  1  and  1963  df: p= < 0.000000000000000222
## Wald test for OBRAT
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = "binomial", data = data1)
## F =  11.55879  on  1  and  1963  df: p= 0.00068779
## Wald test for BLACK
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = "binomial", data = data1)
## F =  11.68361  on  1  and  1963  df: p= 0.0006434
## Wald test for HISPAN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = "binomial", data = data1)
## F =  7.696649  on  1  and  1963  df: p= 0.0055847
## Wald test for LOANPRC
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = "binomial", data = data1)
## F =  10.76933  on  1  and  1963  df: p= 0.0010501

For every one unit change in OBRAT, the log odds of loan approval (versus non loan approval) decreases by 0.0351171.

For every one unit change in LOANPRC, the log odds of loan approval (versus non loan approval) decreases by 0.0165852.

The log odds of loan approval for applicants that meet credit guidelines increases by 3.6946116.

The log odds of loan approval for Black applicants decreases by 0.8176845.

The log odds of loan approval for Hispanic applicants decreases by 0.8582767.

For example, for a black person whose credit history meets guidline (GDLIN = 1), loan amount price is 100 (LOANPRC = 100) and other obligations as a percent of total income is none (OBRAT = 0), the log odds of loan approval is 94.72%

CIs using profiled log-likelihood

##                   2.5 %       97.5 %
## (Intercept)  0.59360475  2.764561689
## GDLIN1       3.28231679  4.126731408
## OBRAT       -0.05537654 -0.014888611
## BLACK1      -1.27852143 -0.339452086
## HISPAN1     -1.44328269 -0.227316096
## LOANPRC     -0.02676923 -0.006824773

CIs using standard errors

##                   2.5 %       97.5 %
## (Intercept)  0.58607300  2.750024092
## GDLIN1       3.27321875  4.116004514
## OBRAT       -0.05536176 -0.014872430
## BLACK1      -1.28654690 -0.348822144
## HISPAN1     -1.46462838 -0.251925037
## LOANPRC     -0.02649064 -0.006679743

Odds ratios only

## (Intercept)      GDLIN1       OBRAT      BLACK1     HISPAN1     LOANPRC 
##   5.3018115  40.2299456   0.9654924   0.4414526   0.4238919   0.9835516

Odds ratios and 95% CI

##                     OR      2.5 %     97.5 %
## (Intercept)  5.3018115  1.8105031 15.8720816
## GDLIN1      40.2299456 26.6374146 61.9750205
## OBRAT        0.9654924  0.9461288  0.9852217
## BLACK1       0.4414526  0.2784487  0.7121604
## HISPAN1      0.4238919  0.2361513  0.7966689
## LOANPRC      0.9835516  0.9735859  0.9931985

Model comparison

## Analysis of Deviance Table
## 
## Model 1: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED + 
##     MALE
## Model 2: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED
## Model 3: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)  
## 1      1961     959.40                       
## 2      1962     959.46 -1  -0.0531  0.81770  
## 3      1963     965.87 -1  -6.4163  0.01131 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Likelihood ratio test
## 
## Model 1: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED + 
##     MALE
## Model 2: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED
## Model 3: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC
##   #Df  LogLik Df  Chisq Pr(>Chisq)  
## 1   8 -479.70                       
## 2   7 -479.73 -1 0.0531    0.81770  
## 3   6 -482.94 -1 6.4163    0.01131 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Probit Regression Full Model

Probit(approve) = \(b_0 + b_1 * GDLIN + b_2 * OBRAT + b_3 * BLACK + b_4 * HISPAN + b_5 * LOANPRC + b_6 * MARRIED + b_7 * MALE\)

## 
## Call:
## glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.9694   0.2361   0.3119   0.3762   2.2711  
## 
## Coefficients:
##              Estimate Std. Error z value             Pr(>|z|)    
## (Intercept)  0.566210   0.309521   1.829             0.067353 .  
## GDLIN1       2.142459   0.121264  17.668 < 0.0000000000000002 ***
## OBRAT       -0.016400   0.005344  -3.069             0.002148 ** 
## BLACK1      -0.425865   0.126958  -3.354             0.000795 ***
## HISPAN1     -0.463475   0.163507  -2.835             0.004588 ** 
## LOANPRC     -0.008409   0.002590  -3.247             0.001166 ** 
## MARRIED1     0.237894   0.096134   2.475             0.013338 *  
## MALE1       -0.033267   0.117944  -0.282             0.777896    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1475.43  on 1968  degrees of freedom
## Residual deviance:  958.84  on 1961  degrees of freedom
## AIC: 974.84
## 
## Number of Fisher Scoring iterations: 6

For every one unit change in OBRAT, the probit odds of loan approval (versus non loan approval) decreases by 0.0163999.

For every one unit change in LOANPRC, the probit odds of loan approval (versus non loan approval) decreases by 0.0084095.

The probit odds of loan approval for applicants that meet credit guidelines increases by 2.1424592.

The probit odds of loan approval for married applicants increases by 0.2378937.

The probit odds of loan approval for Black applicants decreases by 0.4258647.

The probit odds of loan approval for Hispanic applicants decreases by 0.4634752.

For example, for a black married male whose credit history meets guidline (GDLIN = 1), loan amount price is 100 (LOANPRC = 100) and other obligations as a percent of total income is none (OBRAT = 0), the probit odds of loan approval is 95.02%

Statistical Tests for Individual Predictors

## Wald test for GDLIN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## F =  312.1492  on  1  and  1961  df: p= < 0.000000000000000222
## Wald test for OBRAT
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## F =  9.41885  on  1  and  1961  df: p= 0.0021771
## Wald test for BLACK
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## F =  11.25188  on  1  and  1961  df: p= 0.00081057
## Wald test for HISPAN
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## F =  8.034928  on  1  and  1961  df: p= 0.0046354
## Wald test for MALE
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## F =  0.0795587  on  1  and  1961  df: p= 0.77793
## Wald test for LOANPRC
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## F =  10.54337  on  1  and  1961  df: p= 0.0011857
## Wald test for MARRIED
##  in glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED + MALE, family = binomial(link = "probit"), data = data1)
## F =  6.123722  on  1  and  1961  df: p= 0.013422

Probit Regression Reduced Model 1

Probit(approve) = $b_0 + b_1 * GDLIN + b_2 * OBRAT + b_3 * BLACK + b_4 * HISPAN + b_5 * LOANPRC + b_6 * MARRIED $

## 
## Call:
## glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = binomial(link = "probit"), data = data1)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.9709   0.2362   0.3118   0.3775   2.2602  
## 
## Coefficients:
##              Estimate Std. Error z value             Pr(>|z|)    
## (Intercept)  0.541687   0.297610   1.820             0.068740 .  
## GDLIN1       2.143873   0.121184  17.691 < 0.0000000000000002 ***
## OBRAT       -0.016411   0.005344  -3.071             0.002136 ** 
## BLACK1      -0.422677   0.126641  -3.338             0.000845 ***
## HISPAN1     -0.461722   0.163423  -2.825             0.004723 ** 
## LOANPRC     -0.008386   0.002588  -3.240             0.001196 ** 
## MARRIED1     0.228926   0.090475   2.530             0.011397 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1475.43  on 1968  degrees of freedom
## Residual deviance:  958.92  on 1962  degrees of freedom
## AIC: 972.92
## 
## Number of Fisher Scoring iterations: 6
##            Overall
## GDLIN1   17.691026
## OBRAT     3.070619
## BLACK1    3.337594
## HISPAN1   2.825314
## LOANPRC   3.239942
## MARRIED1  2.530275
##          llh      llhNull           G2     McFadden         r2ML 
## -479.4603569 -737.7148785  516.5090432    0.3500736    0.2307356 
##         r2CU 
##    0.4375655

For every one unit change in OBRAT, the probit odds of loan approval (versus non loan approval) decreases by 0.0164109.

For every one unit change in LOANPRC, the probit odds of loan approval (versus non loan approval) decreases by 0.0083864.

The probit odds of loan approval for applicants that meet credit guidelines increases by 2.1438732.

The probit odds of loan approval for married applicants increases by 0.2289263.

The probit odds of loan approval for Black applicants decreases by 0.4226766.

The probit odds of loan approval for Hispanic applicants decreases by 0.4617224.

For example, for a black married person whose credit history meets guidline (GDLIN = 1), loan amount price is 100 (LOANPRC = 100) and other obligations as a percent of total income is none (OBRAT = 0), the probit odds of loan approval is 95.09%

Probit Regression Reduced Model 2

Probit(approve) = $b_0 + b_1 * GDLIN + b_2 * OBRAT + b_3 * BLACK + b_4 * HISPAN + b_5 * LOANPRC $

## 
## Call:
## glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = binomial(link = "probit"), data = data1)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.8976   0.2466   0.3190   0.3719   2.2140  
## 
## Coefficients:
##              Estimate Std. Error z value             Pr(>|z|)    
## (Intercept)  0.708006   0.290189   2.440             0.014695 *  
## GDLIN1       2.135031   0.120676  17.692 < 0.0000000000000002 ***
## OBRAT       -0.017003   0.005340  -3.184             0.001453 ** 
## BLACK1      -0.426583   0.126242  -3.379             0.000727 ***
## HISPAN1     -0.438093   0.163021  -2.687             0.007202 ** 
## LOANPRC     -0.008356   0.002573  -3.248             0.001163 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1475.43  on 1968  degrees of freedom
## Residual deviance:  965.27  on 1963  degrees of freedom
## AIC: 977.27
## 
## Number of Fisher Scoring iterations: 5
##           Overall
## GDLIN1  17.692269
## OBRAT    3.183979
## BLACK1   3.379077
## HISPAN1  2.687345
## LOANPRC  3.247739
##          llh      llhNull           G2     McFadden         r2ML 
## -482.6350057 -737.7148785  510.1597458    0.3457703    0.2282510 
##         r2CU 
##    0.4328537

For every one unit change in OBRAT, the probit odds of loan approval (versus non loan approval) decreases by 0.017003.

For every one unit change in LOANPRC, the probit odds of loan approval (versus non loan approval) decreases by 0.0083562.

The probit odds of loan approval for applicants that meet credit guidelines increases by 2.1350313.

The probit odds of loan approval for Black applicants decreases by 0.4265831.

The probit odds of loan approval for Hispanic applicants decreases by 0.4380932.

For example, for a black person whose credit history meets guidline (GDLIN = 1), loan amount price is 100 (LOANPRC = 100) and other obligations as a percent of total income is none (OBRAT = 0), the probit odds of loan approval is 94.3%

Model comparison

## Analysis of Deviance Table
## 
## Model 1: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED + 
##     MALE
## Model 2: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED
## Model 3: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)  
## 1      1961     958.84                       
## 2      1962     958.92 -1  -0.0799  0.77742  
## 3      1963     965.27 -1  -6.3493  0.01174 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Likelihood ratio test
## 
## Model 1: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED + 
##     MALE
## Model 2: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + MARRIED
## Model 3: APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC
##   #Df  LogLik Df  Chisq Pr(>Chisq)  
## 1   8 -479.42                       
## 2   7 -479.46 -1 0.0799    0.77742  
## 3   6 -482.64 -1 6.3493    0.01174 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Prediction

Logit Model 1

## 
## Call:  glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = "binomial", data = data1)
## 
## Coefficients:
## (Intercept)       GDLIN1        OBRAT       BLACK1      HISPAN1  
##     1.34239      3.72139     -0.03410     -0.81143     -0.89733  
##     LOANPRC     MARRIED1  
##    -0.01677      0.46093  
## 
## Degrees of Freedom: 1968 Total (i.e. Null);  1962 Residual
## Null Deviance:       1475 
## Residual Deviance: 959.5     AIC: 973.5
GDLIN OBRAT BLACK HISPAN MARRIED LOANPRC fit PredictedProb
0 0 1 0 0 77.06418 -0.7616686 0.3182841
0 1 1 0 0 77.06418 -0.7957664 0.3109318
0 2 1 0 0 77.06418 -0.8298643 0.3036738
0 3 1 0 0 77.06418 -0.8639622 0.2965122
0 4 1 0 0 77.06418 -0.8980601 0.2894493
0 5 1 0 0 77.06418 -0.9321579 0.2824871

GDLIN OBRAT BLACK HISPAN MARRIED LOANPRC fit PredictedProb
0 32.38382 1 0 0 2.105000 -0.6085669 0.3523862
0 32.38382 1 0 0 4.681141 -0.6517776 0.3425891
0 32.38382 1 0 0 7.257283 -0.6949883 0.3329243
0 32.38382 1 0 0 9.833424 -0.7381990 0.3233981
0 32.38382 1 0 0 12.409566 -0.7814096 0.3140162
0 32.38382 1 0 0 14.985707 -0.8246203 0.3047838

Logit Model 2

## 
## Call:  glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = "binomial", data = data1)
## 
## Coefficients:
## (Intercept)       GDLIN1        OBRAT       BLACK1      HISPAN1  
##     1.66805      3.69461     -0.03512     -0.81768     -0.85828  
##     LOANPRC  
##    -0.01659  
## 
## Degrees of Freedom: 1968 Total (i.e. Null);  1963 Residual
## Null Deviance:       1475 
## Residual Deviance: 965.9     AIC: 977.9
GDLIN OBRAT BLACK HISPAN LOANPRC fit PredictedProb
0 0 1 0 77.06418 -0.4277603 0.3946613
0 1 1 0 77.06418 -0.4628774 0.3863034
0 2 1 0 77.06418 -0.4979945 0.3780121
0 3 1 0 77.06418 -0.5331116 0.3697914
0 4 1 0 77.06418 -0.5682287 0.3616456
0 5 1 0 77.06418 -0.6033458 0.3535786

GDLIN OBRAT BLACK HISPAN LOANPRC fit PredictedProb
0 32.38382 1 0 2.105000 -0.3217734 0.4202436
0 32.38382 1 0 4.681141 -0.3644992 0.4098709
0 32.38382 1 0 7.257283 -0.4072250 0.3995777
0 32.38382 1 0 9.833424 -0.4499508 0.3893725
0 32.38382 1 0 12.409566 -0.4926766 0.3792632
0 32.38382 1 0 14.985707 -0.5354024 0.3692578

Probit Prediction

Probit Model 1

## 
## Call:  glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC + 
##     MARRIED, family = binomial(link = "probit"), data = data1)
## 
## Coefficients:
## (Intercept)       GDLIN1        OBRAT       BLACK1      HISPAN1  
##    0.541687     2.143873    -0.016411    -0.422677    -0.461722  
##     LOANPRC     MARRIED1  
##   -0.008386     0.228926  
## 
## Degrees of Freedom: 1968 Total (i.e. Null);  1962 Residual
## Null Deviance:       1475 
## Residual Deviance: 958.9     AIC: 972.9
GDLIN OBRAT BLACK HISPAN MARRIED LOANPRC fit se.fit
0 0 1 0 0 77.06418 0.2989992 0.0861630
0 1 1 0 0 77.06418 0.2933267 0.0839904
0 2 1 0 0 77.06418 0.2877046 0.0818437
0 3 1 0 0 77.06418 0.2821340 0.0797245
0 4 1 0 0 77.06418 0.2766158 0.0776346
0 5 1 0 0 77.06418 0.2711510 0.0755755

GDLIN OBRAT BLACK HISPAN MARRIED LOANPRC
0 32.38382 1 0 0 2.105000
0 32.38382 1 0 0 4.681141
0 32.38382 1 0 0 7.257283
0 32.38382 1 0 0 9.833424
0 32.38382 1 0 0 12.409566
0 32.38382 1 0 0 14.985707

Probit Model 2

## 
## Call:  glm(formula = APPROVE ~ GDLIN + OBRAT + BLACK + HISPAN + LOANPRC, 
##     family = binomial(link = "probit"), data = data1)
## 
## Coefficients:
## (Intercept)       GDLIN1        OBRAT       BLACK1      HISPAN1  
##    0.708006     2.135031    -0.017003    -0.426583    -0.438093  
##     LOANPRC  
##   -0.008356  
## 
## Degrees of Freedom: 1968 Total (i.e. Null);  1963 Residual
## Null Deviance:       1475 
## Residual Deviance: 965.3     AIC: 977.3
GDLIN OBRAT BLACK HISPAN LOANPRC fit
0 0 1 0 77.06418 0.3584733
0 1 1 0 77.06418 0.3521414
0 2 1 0 77.06418 0.3458502
0 3 1 0 77.06418 0.3396013
0 4 1 0 77.06418 0.3333962
0 5 1 0 77.06418 0.3272363

GDLIN OBRAT BLACK HISPAN LOANPRC fit
0 32.38382 1 0 2.105000 0.3871373
0 32.38382 1 0 4.681141 0.3789214
0 32.38382 1 0 7.257283 0.3707598
0 32.38382 1 0 9.833424 0.3626560
0 32.38382 1 0 12.409566 0.3546132
0 32.38382 1 0 14.985707 0.3466347